Problem 103 - A Graphical Approach to College Algebra

Question

Solve each problem involving rate of work. Mrs. Schmulen is a high school mathematics teacher. She can grade a set of chapter tests in 5 hours working alone. If her student teacher Elwyn helps her, it will take 3 hours to grade the tests. How long would it take Elwyn to grade the tests if he worked alone?

Step-by-Step Solution

Answer

Elwyn can grade the tests alone in 7.5 hours.

Step 1: Determine Mrs. Schmulen's Rate

Mrs. Schmulen can finish grading 1 set of tests in 5 hours. Therefore, her rate of work is \(\frac{1}{5}\) of the test set per hour.

Step 2: Calculate the Combined Work Rate

When Mrs. Schmulen and Elwyn work together, they finish grading in 3 hours. Hence, their combined work rate is \(\frac{1}{3}\) of the test set per hour.

Step 3: Express Elwyn's Rate

Let Elwyn's rate of work be \(\frac{1}{x}\) of the test set per hour. The equation for their combined rate is \(\frac{1}{5} + \frac{1}{x} = \frac{1}{3}\).

Step 4: Solve for Elwyn's Rate

To find \(x\), solve the equation:\[ \frac{1}{5} + \frac{1}{x} = \frac{1}{3} \]Subtract \(\frac{1}{5}\) from both sides:\[ \frac{1}{x} = \frac{1}{3} - \frac{1}{5} \]Find a common denominator, which is 15:\[ \frac{1}{3} = \frac{5}{15} \quad \text{and} \quad \frac{1}{5} = \frac{3}{15} \]Then:\[ \frac{1}{x} = \frac{5}{15} - \frac{3}{15} = \frac{2}{15} \]

Step 5: Calculate Elwyn's Time to Complete the Task

Solving \(\frac{1}{x} = \frac{2}{15}\) yields:\[ x = \frac{15}{2} = 7.5 \]So, Elwyn would take 7.5 hours to grade the tests alone.

Key Concepts

Work Rate

The concept of work rate is essential in understanding how tasks are completed over time. In rate of work problems, we consider how much of a task can be done in a certain period. If someone can complete a task in a specific amount of time, their work rate is expressed as the fraction of the task done per unit time.
The formula for work rate is:

• Work Rate = \( \frac{1}{\text{Time}} \)

In the exercise, Mrs. Schmulen can complete one set of tests in 5 hours, so her work rate is \( \frac{1}{5} \) of the test set per hour. Combining work rates helps determine the overall efficiency when people work together. Understanding individual work rates allows you to form equations representing how long it would take to complete tasks alone or with others.
For example, when calculating the combined work rate of Mrs. Schmulen and Elwyn, knowing the rate at which each can work helps derive the equation used to solve for Elwyn's rate.

Algebraic Equations

Algebraic equations are tools used to express relationships between quantities. They are crucial for solving rate of work problems where multiple variables interact. In the exercise, variables represent individual work rates, and algebra forms a bridge connecting these rates.
Here, we let Elwyn's work rate be \( \frac{1}{x} \) of the test set per hour. The combined rate with Mrs. Schmulen is established as an equation:

• \( \frac{1}{5} + \frac{1}{x} = \frac{1}{3} \)

This equation expresses how their combined efficiency equals \( \frac{1}{3} \) when working together. Solving it requires step-by-step manipulation, like subtracting and finding common denominators, to isolate the unknown variable \( x \).
Such equations reveal the power of algebra in translating word problems into solvable mathematical sentences, guiding us to the solution.

Problem Solving

Problem solving in rate of work scenarios involves analyzing given information, translating it into mathematical expressions, and finding solutions. It's a methodical process that draws heavily on understanding work rates and algebra.
First, analyze the problem to capture the key information, like individual and joint work times. Next, identify what you need to find—in this case, Elwyn's time when working alone.
Construct the equation using known work rates and solve for the unknown. This requires:

• Subtracting known rates from their combined rate.
• Utilizing algebraic techniques to isolate and calculate the unknown rate.
• Interpreting the result — here, it shows Elwyn needs 7.5 hours alone to finish the task.
  

Every step is aimed at breaking down the problem, giving clarity and structure to finding the solution.